Quantum adiabatic theorem for unbounded Hamiltonians with a cutoff and its application to superconducting circuits

We present a new quantum adiabatic theorem that allows one to rigorously bound the adiabatic timescale for a variety of systems, including those described by originally unbounded Hamiltonians that are made finite-dimensional by a cutoff. Our bound is geared towards the qubit approximation of superconducting circuits and presents a sufficient condition for remaining within the 2n-dimensional qubit subspace of a circuit model of n qubits. The novelty of this adiabatic theorem is that, unlike previous rigorous results, it does not contain 2n as a factor in the adiabatic timescale, and it allows one to obtain an expression for the adiabatic timescale independent of the cutoff of the infinite-dimensional Hilbert space of the circuit Hamiltonian. As an application, we present an explicit dependence of this timescale on circuit parameters for a superconducting flux qubit and demonstrate that leakage out of the qubit subspace is inevitable as the tunnelling barrier is raised towards the end of a quantum anneal. We also discuss a method of obtaining a 2n×2n effective Hamiltonian that best approximates the true dynamics induced by slowly changing circuit control parameters. This article is part of the theme issue ‘Quantum annealing and computation: challenges and perspectives’.


Introduction
The quantum adiabatic theorem is now more than 100 years old, dating back to Einstein [1] and Ehrenfest [2]. Yet, it still continues to inspire new interest and results, in large part owing to its central role in adiabatic quantum computation and quantum annealing, where it can be viewed as providing a sufficient condition for the solution of hard computational problems via adiabatic quantum evolutions [3][4][5].
Consider a closed quantum system evolving for a total time t f subject to the Hamiltonian H(t). Defining the rescaled (dimensionless) time s = t/t f , the evolution is governed by the unitary operator U tot (s) which is the solution of 1 U tot (s) = −it f H(s)U tot (s), U tot (0) = I, s ∈ [0, 1]. (1.1) In this work, we assume that the Hamiltonian H(s) ≡ H Λ (s) is defined as an operator on a finite-dimensional Hilbert space H of dimension Λ, but it is obtained via discretization of an unbounded Hamiltonian H ∞ over an infinite-dimensional Hilbert space. By unbounded we mean that the energy expectation value ψ|H ∞ |ψ can be arbitrarily large for an appropriate choice of |ψ within the domain where H ∞ is defined. We will not, however, work with that unbounded Hamiltonian directly, so all our proofs will use the properties of finite-dimensional Hamiltonians, e.g. that the solution to the Schrödinger equation exists and the spectrum of H Λ (s) comprises Λ discrete (possibly degenerate) eigenvalues. In particular, we will not assume that the limit as Λ → ∞ of any of the quantities appearing in our results exists. The dimension Λ < ∞ is what throughout this work we call the cutoff. We will outline a path to proving a somewhat weaker result for unbounded Hamiltonians H ∞ themselves, but leave a rigorous proof for future work.
Let P(s) be a finite-rank projection on the low-energy subspace of H(s), i.e. the (continuous-in-s) subspace spanned by the eigenvectors with the lowest d(s) eigenvalues. A unitary operator U ad (s) can be constructed that preserves this subspace, i.e. P(s) = U ad (s)P(0)U † ad (s). (1. 2) The adiabatic theorem is essentially the statement that there exists U ad such that the following holds: 2 [U ad (s) − U tot (s)]P(0) ≤ θ t f ≡ b, (1.3) where θ is a constant that does not depend on the final time t f but typically (though not always [7,8]) depends on the minimum eigenvalue gap of H(s) between P(s)H and Q(s)H, where Q = I − P. Since the right-hand side (r.h.s.) represents the deviation from adiabaticity, henceforth we refer to b as the 'diabatic evolution bound' and to θ as the 'adiabatic timescale'. The total evolution time is adiabatic if it satisfies t f θ . Thus, the system evolves adiabatically (diabatically) if the diabatic evolution bound is small (large).
This version of the adiabatic theorem amounts to finding an expression for U ad that contains information about the dynamic and geometric phase acquired along the evolution, and can be found in the book [8] for unbounded operators. Note that typical textbook expressions (e.g. [9]) just bound the overlap between U ad (1)|ψ(0) and the final state U tot (1)|ψ(0) , where |ψ(0) is the lowest eigenstate of H(0). Instead, we consider any initial state |ψ(0) ∈ P(0)H, not just the ground state, and also compute the total phase. This is also more flexible in that, in fact, the projector P can single out any subspace of eigenstates of H (not necessarily the lowest), which may or may not be degenerate.
Techniques exist to improve the bound to γ k /t k f for integers k > 1. This is done by requiring the time-dependent Hamiltonian to have vanishing derivatives up to order k at the initial and final into that Hilbert space and the matrix H eff such that the solution of the Schrödinger equation u (s) = −it f H eff (s)u(s) with u(0) = I is close to the true evolution due to the same adiabatic theorem stated above: u(s) − V(s)U tot (s)V † (s) ≤ b. (1.4) We apply our results to circuits of superconducting flux qubits [29,30], of the type used, for example, in quantum annealing [31][32][33]. Quantum annealing (reviewed in [5,[34][35][36]) is a field that primarily studies heuristic quantum algorithms for optimization, best suited to running on analogue quantum devices. In the qubit language, the quantum annealer is typically initialized in a uniform superposition state that is the ground state of a transverse field Hamiltonian. Over the course of the algorithm, the strength of the transverse field is gradually decreased while simultaneously the strength of the interactions encoding the optimization problem of interest is gradually increased, guiding the quantum evolution towards the ground state that encodes an optimal solution. In the context of superconducting devices, the qubits used for this, with frequency ω q , are described by a circuit model (which includes capacitors, Josephson junctions etc.), characterized by the capacitive energy E C and the Josephson junction energy E J E C . We express the plasma frequency ω pl (s) and the residual transverse field ω q δ at the end of the anneal via the circuit parameters E J and E C and the schedule of the control fluxes. We obtain a bound for the adiabatic timescale θ in equation (1.3), ω q θ = O(ω q /(ω pl (1)δ))(ln(ω pl (1)/(ω q δ))) −1 , while applying the existing analytically tractable form of the adiabatic theorem [25] yields ω q θ = Θ(Λ), 4 which diverges with the cutoff. We also check that for finite H the existing form [25] gives a result that is consistent with our bound, namely ω q θ = O(ω q /(ω pl (1)δ)). For these expressions written in terms of E J and E C see §5b. Thus, our results include the first non-diverging expression for the adiabatic timescale in the case of unbounded Hamiltonians, as well as a new practical application of existing rigorous forms of the adiabatic theorem. The structure of the rest of this paper is as follows. We provide detailed definitions required to state our result, as well as compare it with previous work, in §2. The paper is written in a way that allows the reader to skip the proof that follows this section and move on to applications in §5. The proof is given in two parts: a short argument for obtaining an O(1/t f ) bound in §3 and a lengthier part in §4 in which we compute the constant θ. The application to flux qubits can be found in §5, which is also separated into results and a proof that can be skipped. We give the definition of the effective (qubit) Hamiltonian in §6, along with a discussion of how the adiabatic theorem bounds we obtained apply in the effective Hamiltonian setting. Sections 5 and 6 are independent of each other. We conclude in §7. Additional calculations in support of the flux qubit analysis are presented in appendix A, and a proof of the intertwining relation is given in appendix B.

Adiabatic and diabatic evolution (a) Previous work
To set the stage for our results on the adiabatic theorem, we first briefly review key earlier results. We note that, unlike these earlier works, we will provide an explicit expression for the adiabatic timescale, which does not diverge with the cutoff of the Hamiltonian in most relevant examples and is ready to be used both analytically and numerically. This is an important novel aspect of our contribution to the topic.
Such a ready-to-use result was obtained for finite-dimensional (bounded) Hamiltonians by Jansen, Ruskai and Seiler (JRS), and our results closely follow their work. They proved several bounds, including the following [ Then The direct dependence on H and H is the crucial one from our perspective, and the one we avoid in this work. Indeed, these norms diverge with the cutoff for a time-dependent harmonic oscillator or the hydrogen atom, for example.
The adiabatic timescale that is harder to use analytically and numerically can be found in [8, eqn (2.2)]: and Γ is a contour around the part of the spectrum corresponding to PH. In what follows we give a simplified non-rigorous summary of the arguments used in [8] to prove that θ < ∞. The boundedness of the norm of F and its derivative can be traced down to an assumption, where we have kept an energy scale ε to match the dimensions, but ε = 1 is usually taken in the mathematical literature. The smallest such constant, C ε = H L(D,H) , is actually the definition of the operator norm for unbounded Hamiltonians with a domain D. The space D is equipped with, besides the usual state norm ψ H inherited from H, a different state norm ψ D than H, called the graph norm: for some Hamiltonian H 0 (which we take to be equal to H for a tighter bound) and some arbitrary energy scale ε. The operator norms are now computed with respect to the spaces they map between: At the cost of the small increase in norm of the resolvent, we have obtained a finite number C ε in place of the norm of the unbounded operator. Using this idea, in [8] it is proved that θ < ∞. Note that for finite-dimensional systems the assumption (2.4) can also be written as 5  [7], §5), while mainly focusing on gapless bounded Hamiltonians, discussed the adiabatic theorem for unbounded gapless Hamiltonians. They required that both the resolvent R(z = i, s) and H(s)R (z = i, s) be bounded. Essentially the same assumption was made by Abou Salem [37, §2] in the context of non-normal generators.
Recent work [38,39] presents a refinement of the adiabatic theorem for a different case of diverging H that comes from the thermodynamic limit of the size of a many-body spin system. While the authors do not present an explicit form for θ, we believe that their methods provide an alternative way of removing the dimension d of the subspace PH, and in fact any dependence on the system size, from the bound on local observables.

(b) Adiabatic intertwiner
Following Kato [20], we define an approximate evolution in the full Hilbert space H: where U ad is called the adiabatic intertwiner and the (dimensionless) adiabatic Hamiltonian is (2.10) Note that both H ad and U ad are t f -dependent. Here P(s) is a finite-rank projection on the lowenergy subspace of H(s) (i.e. the continuous-in-s subspace spanned by the eigenvectors with the lowest d(s) eigenvalues 6 ). A property of this approximation is that the low-energy subspace is preserved: U ad (s)P 0 = P(s)U ad (s), (2.11) where here and henceforth we denote P(0) by P 0 and drop the s time-argument from P(s) where possible. The proof of this intertwining property is well known and has been given many times in various forms and subject to various generalizations; see e.g. [22,26,37,40,41] as well as our appendix B. The idea (due to Kato [20], who presented the original proof; see his eqn (22)) is to show that both sides solve the same initial value problem, i.e. equality holds at s = 0, and they satisfy the same differential equation after differentiating by s. The latter can be shown using equations (2.12) and (2.13) below. The operator P has the following useful properties. Since P 2 = P, we have Multiplying by P on the right and letting Q ≡ I − P, we obtain QP P = P P, i.e.
PP P = 0 and QP Q = 0, (2.13) where the proof of QP Q = 0 is similar. Thus P is block-off-diagonal: (2.14) We also note that for a spatially local system the generator related to i[P , P] is approximately a sum of local terms [42]. This approximation is known as a quasi-adiabatic continuation [43], though we will not discuss locality in this work.

(c) Bounds on states and physical observables
We would like to bound certain physical observables via the quantity b defined in equation (1.3). Since b bounds the difference between the actual and adiabatic evolution, we refer to b as the 'diabatic evolution bound'.
We note that Kato's adiabatic theorem [20] established that for bounded Hamiltonians, the quantity [U ad (s) − U tot (s)]P 0 tends to zero as 1/t f , but it will still take us most of the rest of this paper to arrive at the point where we can state with conviction that the bound in equation (1.3) does not diverge with the cutoff. This will require extra assumptions; indeed, there are contrived unbounded Hamiltonians for which Kato's quantity is arbitrarily large for any finite evolution time t f .
Note that by using unitary invariance we can rewrite equation Consider an initial state |φ in the low-energy subspace (P 0 |φ = |φ ). We wish to compare the evolution generated by U tot with that generated by U ad . Dropping the s time-argument from the U's, the difference in the resulting final states is We use this quantity because we would like to describe the error in both the amplitude and the acquired phase of the wave function.

(ii) Bound on leakage
If we are just interested in the leakage from the low-lying subspace, it can be expressed as To prove this, note that Therefore, One of the immediate consequences is that measuring Z (or any other unit-norm observable) on one qubit in an n-qubit system after the evolution can be described by an approximate evolution U ad to within an error of 2b + b 2 in the expectation value.

(iv) Bound on the JRS quantity
The quantity appearing in the JRS bound (2.1) satisfies where in the last equality we used Q 0 = I − P 0 and added/subtracted P 0 U † ad U tot P 0 . Using the definition of x (equation (2.15)), we can write so that equation (2.23) becomes where the second equality holds since Q 0 U † ad U tot x † P 0 and P 0 xQ 0 are two opposite off-diagonal blocks and their eigenvalues do not mix, and the last equality follows from the unitary invariance of the operator norm.
We proceed to explicitly express the bound b in the next subsection.

(d) Statement of the theorem
Collecting the definitions of the previous sections, we present our main result.  1. Assume that the initial state |φ ∈ P(0) ≡ P 0 . Then the adiabatic intertwiner U ad (the solution of equation (2.9)) satisfies the following bounds on its difference from the true evolution U tot :

27)
where b = θ/t f with θ given by  (2.25). The new aspect of theorem 2.1 is the value of the bound θ, which does not involve H or higher derivatives that may diverge with the cutoff used to define H(s). Moreover, PH Q gives a tighter bound than the H L(D,H) that would have been obtained from direct translation of the adiabatic theorem for unbounded Hamiltonians given in [8]. Indeed, In terms of c 0 and c 1 , PH Q ≤ c 0 + c 1 r(s) 2 /4. When the above inequalities are tight, our bound would match the one that could in principle be obtained from [8]. However, in many relevant cases, such as a harmonic oscillator with small time-dependent anharmonicity, PH Q is parametrically less than the r.h.s. We also find the form of PH Q to be more insightful than H L(D,H) . Since the constants c 0 and c 1 depend on the choice of the constant energy offset, we chose zero energy to lie in the middle of the eigenvalues corresponding to PH. We note that for bounded H the assumption (2.26) is automatically satisfied with c 1 (s) = 0 and c 0 (s) = H 2 , since H 2 − H 2 I ≤ 0 (a negative operator) by definition of the operator norm. Using this, we can reduce equation ( We see that, though our new adiabatic timescale has slightly larger numerical coefficients, the projected form of the operators can provide a qualitative improvement over the JRS result. 7 Note that we can also write a bound that is free of the dimension d if the second option for τ in equation (2.29) is smaller than the first.

Diabatic evolution bound
We will calculate a diabatic evolution bound b on the quantity in equation (1.3) for some s * ∈ [0, 1]: We would like to express f (s * ) via f (s) as Recalling that U tot satisfies equation (1.1) and U ad satisfies equation (2.9), the derivative is Using this in equation (3.4), we obtain the desired O(1/t f ) scaling: where by using equation (2.11) we simplified one term in the commutator as P 0 U † ad P = P 0 U † ad , and also by using equation (2.13) we have P 0 U † ad P P = U † ad PP P = 0, so that the other term with P P in the commutator vanishes. Plugging this back into equation (3.3), we get Using P 0 U † ad = U † ad P throughout, this results in the following bound on the quantity in equation (3.1) we set out to bound: The adiabatic timescale θ given here is not particularly useful in its present form. So we next set out to find bounds on each of the quantities involved. Our goal will be to bound everything in terms of block-off-diagonal elements of H and its derivatives, i.e. terms of the form PHQ , PH Q etc.

Bounds via the resolvent formalism
Some of the material in this section closely follows Jansen et al. (JRS) [25], adjusted for clarity for our purposes. We start from the well-known resolvent formula and then develop various intermediate bounds we need for the final result.

(a) Twiddled operators
If Γ is a positively oriented loop in the complex plane encircling the spectrum associated with an orthogonal eigenprojection P of a Hermitian operator H, then [44] where (H − z) −1 is known as the resolvent. Using this, it was shown in lemma 2 of [25] that for every operator X there is a solutionX to equation (3.5) if the eigenvalues in P are separated by a gap in H. This solution is written as follows in terms of contour integrals involving the double resolvent: 9 where the contour Γ again encircles the portion of the spectrum within P. HereX is block-offdiagonal. The twiddle operation was introduced in [ Also note thatX is block-off-diagonal [25], i.e.
We also know that P is block-off-diagonal, so by definition (equation (3.5)) But the tilde operation depends only on the block-off-diagonal elements of H , so that which implies that as long as this quantity is bounded, P is as well: P = (PH Q + QH P) ∼ .

(c) Bound onX
Suppose that the spectrum of H(s) (its eigenvalues {E i (s)}) restricted to P(s) consists of d(s) eigenvalues (each possibly degenerate, with crossing permitted) separated by a gap of 2 (s) from the rest of the spectrum of H(s). That is, d(s) ≤ d, the dimension of the low-energy subspace. Under these assumptions, JRS proved the following bound in their lemma 7: We will also use an alternative bound that did not appear in [25]. We start with for z on the contour Γ in equation (4.2), illustrated in figure 1. This contour is of length 2r(s) + 2π (s) where r is the spectral diameter of PH with respect to H. Since P(s) is a spectrum projector, PH has a basis of eigenvectors of H(s) with eigenvalues λ P i , and we can define So, bounding the solutionX(s) from equation (4.2) directly results in This new bound can be tighter than equation ( Here, τ roughly means the adiabatic timescale. The bound (4.12) can be seen as one of the main reasons for introducing the twiddle operation. We will use it repeatedly below. We will omit the s-dependence of τ andX whenever possible in what follows. Note that if Y is any operator that commutes with H, then by equation (4.2) we haveXY = (XY) ∼ and YX = (YX) ∼ . Therefore Likewise, using equations (4.3), (4.4) and (4.12) we can remove a twiddle under the operator norm for the price of a factor of τ while inserting P and Q at will: (4.14)

(d) Combining everything into the diabatic evolution bound
We now combine the various intermediate results above to bound the r.h.s. of equation (3.12).
Now, using [P, H] = 0 and PP P = 0, note that Also, PP = (PP ) † = P P (since P and P are Hermitian), so by using equation (4.3) we obtain We multiply equation (4.7) from the left by P to give where we used equation (4.3). Therefore, using X ≤ τ X again, we find that We have nearly achieved the goal of expressing the diabatic evolution bound in terms of blockoff-diagonal elements of H and its derivatives. The last term is not yet in this form and will require the development of additional tools, which we do next.

(e) Derivative of the resolvent formula
To take derivatives of the twiddled expressions, we need to differentiate the resolvent R(z, s) = (H(s) − z) −1 . By differentiating the identity (H(s) − z))R(z, s) = I we obtain We will apply the derivative formula to our derivation. For example, using equation (4.2) we obtain and hence taking the derivative results in To bound this expression, we need to prove one more fact.

(f) Fact about a triple resolvent
We will need to analyse expressions of the form which we will use with A, B = H for the norm of P and A, B = H , P for the bound on P ∼ above. That is, JRS proved a bound on F(A, B). Since F(A, B) has both diagonal and off-diagonal blocks, they found the bound for each block. We review their proof below, starting from a useful expression for the triple resolvent.
Consider the commutator with the Hamiltonian: where we have inserted z since it is not an operator and therefore commutes with the other term, and where the second equality follows from equation (4.2). Let us denote the off-diagonal block projection by o(X) = PXQ + QXP = [P, (P − Q)X]. Note that P and Q commute with H, so when we apply [P, (P − Q) ·] to both sides of the above equation, we get, after some simple algebra,  For the block-diagonal part, we need to apply a different strategy. By pole integrations identical to those in [25], which only require that there be a finite number of eigenvalues inside the lowenergy subspace, we can prove that (4.31) Combining the last two results, we finally obtain (the same as equation (13) in [25]) Now, using equations (4.2), (4.6) and (4.22), we can express P as It then follows from equation (4.32) that (g) Bounding the last term in the diabatic evolution bound We are interested in bounding the last term in equation (4.21), which by using equation (4.26) we can write as Recall that P = −H ∼ (equation (4.6)), so that Repeatedly using the fact that twiddled operators are block-off-diagonal and using equation (4.14), we find that where in the last inequality we used equation (4.2) and the fact that both P and P are Hermitian to write P ∼ P = (PP ∼ ) † = PP ∼ . Similarly, where in the second equality we used PX =XQ (equation (4.4b)). The remaining terms in equation (4.37) are similarly bounded: and Combining these bounds yields To deal with the two terms that still contain ∼( PP ∼ H and PH ∼ H ), we have no choice but to use the constants c 0 and c 1 introduced in §2: We use this assumption as follows. First, it implies that Hence, upon taking norms of both sides, where in the first equality we used A 2 = AA † and in the last equality we made use ofXY = (XY) ∼ when [Y, H] = 0 and then applied equation (4.14). Similarly, using P = −H ∼ , The quantity PH HQ appearing for k = 1 is usually well behaved with Λ, as we will see in examples in §5. In case it is not, we need to take a step back and recall that we obtained it via the bound P(H H) ∼ Q ≤ τ PH HQ , which follows from equation (4.14). We thus consider undoing this bound and replacing τ PH HQ with P(H H) ∼ Q . Using the definition of the ∼ operation (equation (4.2)), where to obtain the second equality we used (P/(2π i) The choice of zero energy right in the middle of the eigenvalues corresponding to PH ensures that |z| ≤ r/2 + for z ∈ Γ (figure 1). Using this fact along with equation (4.11) then results in the bound Alternatively, a slight adjustment to the derivation in [25] gives We are now ready to write down the diabatic evolution bound in its final form, by combining equations (1.3), (3.1), (4.21) and (4.50): where the expression for θ coincides with the one in equation (2.28) and hence serves as the end of the proof of theorem 2.1. It is worth recalling here also that τ contains a gap dependence via equation (2.29). Note that despite appearances due to the block-off-diagonal form of this bound, all of the terms involved can be bounded by norms of some d P × d P matrices (where d P = rank(P)): where the inequalities follow by writing (for any Hermitian operator A) PAQ = max |v ,|w v|PAQ|w ≤ max |v ,|w v|PA|w = PA and PA 2 = PA(PA) † ≤ PA 2 P , so that PAQ 2 ≤ PA 2 P . Before we proceed, let us comment briefly on a physical consequence of the bound [U ad (s * ) − U tot (s * )]P 0 ≤ θ/t f that we have just proven (equation (4.51)). In §2c(iii), we gave a bound on the difference in expectation value of an observable O between the exact and the adiabatic evolution. Suppose that O is a unit-norm observable such as the Pauli matrix σ z ≡ Z or σ x ≡ X; measuring Z on a single qubit in an n-qubit system is a standard 'computational basis' measurement. For this example, equation (2.20) then becomes This means that a measurement of Z at t f has an expectation value that-provided θ/t f 1-is well described by an expectation value computed from the evolution U ad that never leaves the low-energy subspace, which is the qubit subspace. The error between the two is given by the bound above. In §6, we discuss the effective Hamiltonian (a qubit Hamiltonian for this example) generating this approximate evolution in more detail, with the aim of providing a recipe for numerical simulations of qubit Hamiltonians that can predict the outcomes of superconducting circuit experiments.

Examples
We consider examples motivated by adiabatic quantum computing and quantum annealing with flux qubits [33,[45][46][47][48]. We first discuss inductively coupled flux qubits in terms of generic circuit Hamiltonians. We use theorem 2.1 to derive general bounds on the deviation between the actual evolution described by these circuit Hamiltonians and the evolution in the desired low-energy subspace defined by P. Next we discuss specific models of single flux qubits, for which we can explicitly exhibit the dependence of our bounds on the circuit parameters.

(a) Application to coupled flux qubits
An interesting example is the circuit Hamiltonian describing inductively coupled superconducting flux qubits [49]: wherep i andx i are canonically conjugate momentum and position operators, respectively. The remaining quantities are scalar control parameters: the ϕ i are control fluxes, the M ij are matrix elements of the mutual inductance matrix, and the B i are barrier heights depending on more control fluxes [30]. A simplified circuit described by this equation is shown in figure 2. For notational simplicity, we drop the hat (operator) notation below.
The Hamiltonian H flux (s) is defined over an infinite-dimensional Hilbert space and is unbounded: H flux (s) = max |v v|H flux (s)|v is infinite for |v maximized over a typical Hilbert space. One such space can be defined by choosing and considering eigenvectors |v = i |n i of this collection of harmonic oscillators. Clearly, in some contexts in physics, arbitrarily high n i will appear as a physical state, which would lead to arbitrarily large v|p 2 i |v , v|x 2 i |v , v|H 0 |v and v|H flux (s)|v . Indeed, the operators involved would normally be referred to as unbounded. We note that in the definition of the norm · L(D,H) [8] discussed in §2a, these operators are bounded with respect to the Hamiltonian. We choose instead to impose a cutoff on the Hamiltonian directly. This allows us to make comparisons with the JRS result, which requires a finite-dimensional Hamiltonian. Consider a projector P Λ on states with all n i ≤ Λ, and for any operator O on the original infinite-dimensional Hilbert space define O Λ as the finite-dimensional matrix that is the P Λ block of P Λ O Λ P Λ . Now, using the standard definition of the norm for finite-dimensional matrices, we can get and H Λ flux (s) = Θ(Λ). Below we will omit the superscript Λ, but all the expressions that follow are understood to hold in this finite-dimensional space.

(i) Constant mutual inductance matrix
We first consider the case where M ij (s) = M ij . As we shall see, in this case H does not grow with the cutoff, H 2 ≤ c 0 is sufficient, and previously developed bounds such as that of JRS will not depend on the cutoff either, although recall that by corollary 2.2 we can obtain a tighter bound.
The derivative is and we note that where as long as B i (s) and ϕ i (s) are smooth functions of s, then c 0 (s) is finite, does not depend on the cutoff Λ and has dimensions of energy, The final error upper bound (equation (2.28)) simplifies to Now, since in this example H (s) is finite and Λ-independent for all s, in fact the projection P is not necessary and known bounds are already Λ-independent. Indeed, the JRS bound for θ(s * ) quoted in equation (2.1) is clearly Λ-independent for the present example (recall corollary 2.2)). Thus, in the next subsection, we consider an example where H (s) diverges with Λ.
(ii) Time-dependent mutual inductance matrix Generally, to implement a standard adiabatic quantum computing or quantum annealing protocol, the mutual inductance matrix M ij cannot be constant (e.g. see [46]). Thus we consider a second example of a circuit Hamiltonian of superconducting flux qubits, which is more appropriate for both quantum annealing and our purpose of demonstrating the case of unbounded Hamiltonians with cutoff. Consider the Hamiltonian in equation (5.1) and its derivative The term M ij (s)x i x j , containing the derivative of the time-dependent mutual inductance matrix, now grows arbitrarily large in norm with Λ because of the x i x j terms (recall that the x i are operators), so that the JRS version of the adiabatic theorem (equation (2.1)) has an adiabatic timescale that is arbitrarily large in Λ and we need to resort to theorem 2.1. Note that M ij (s) is always a positive matrix. Denote its lowest eigenvalue by l = min λ M . Then we can bound Note also that We now add a (positive) p 2 term and add and subtract the cos term to complete the Hamiltonian: Bounding the last term in the same way as the first two, we obtain Thus the constants we defined in the general notation of equation ( The final numerator in the diabatic evolution bound (equation (2.28)) becomes

(b) Adiabatic timescale via superconducting qubit circuit parameters
The bounds above are stated in terms of the circuit parameters B i and M ij but are too abstract to be practically useful. In this subsection we consider more specific models and arrive at practically useful bounds which also illustrate the utility of our approach for dealing with unbounded operators with a cutoff. We consider two types of flux qubit circuit Hamiltonians: and As we explain below, H CJJ describes a compound Josephson junction (CJJ) rf SQUID qubit [31], while H CSFQ describes a capacitively shunted flux qubit (CSFQ) [32]; H CSFQ can be obtained by analysing the circuit displayed in figure 3. Note that in the notation of equation (  canonically conjugate operatorsn (charge stored in the capacitor C) andφ (flux threading the circuit) are identified withp andx, respectively, and that in the transmon case E L = E α = 0 [50]. 10 The quadratic self-inductance term E L (φ − f ) 2 is responsible for the divergence of H CJJ with the cutoff Λ, just like the time-dependent mutual inductance in equation (5.1). Thus, the JRS adiabatic theorem once again provides an unphysical dependence on the cutoff and the bound we derived in equation (5.16) can be used instead. The adiabatic timescale depends on the choice of schedules for the controls b and f . To illustrate what enters this choice, we first explain how H CJJ can be reduced to an effective qubit Hamiltonian. We would like to stress that we only need the qubit approximation for the schedule choice; the adiabatic timescale we find is a property beyond the qubit approximation, and the approximation itself is not used any more after the schedule is set. Before presenting the result for CJJ qubits, we borrow the same set of tools to find the effective qubit Hamiltonian and explicitly compute our bounds for the capacitively shunted flux qubit described by a simpler Hamiltonian H CSFQ, sin where we retain just one of the trigonometric terms: Note that the derivatives of H CSFQ and H CSFQ, sin do not grow in norm with the cutoff Λ, so in this case the JRS adiabatic theorem provides a useful baseline, but as explained below we will obtain a somewhat tighter bound. The quantities b ≥ 1 and f ≥ 0 are time-dependent controls that can be chosen at will. Ideally, we would like the effective qubit Hamiltonian ( §6) to match a desired quantum annealing 'schedule' ω q ((1 − s)X + sZ) where s = t/t f is the dimensionless time. However, in practice, for calibration of the annealing schedule an approximate method for choosing b(s) and f (s) is used instead. Here we will also follow this approximate method for simplicity; thus we will not know the true effective qubit Hamiltonian H eff the schedule is implementing, but we will be able to accurately bound the error of that qubit description. This is in line with our goal of providing a useful theoretical result to guide current experiments with superconducting circuits: the error would characterize, for instance, the leakage to the non-qubit states for fast anneals. The true effective Hamiltonian H eff , and correspondingly a precise method for choosing b(s) and f (s), can be found straightforwardly in a numerical simulation, which we leave for future work.
The approximate method is as follows.  . For various target Hamiltonians between +ω q Z and −ω q Z, the anneal paths in the parameter space (b(s), f (s)) occupy the white triangle. The yellow triangle indicates the range of applicability of the qubit approximation for anneals with t f θ (s * ). The splittings ω q and ω q δ are obtained at zero bias at the beginning and the end of the anneal, respectively. Maximum bias also yields ω q at the end of the anneal. The plasma frequency ω pl is the frequency of each well, and it increases throughout the anneal towards the value ω pl (s * ) that enters θ (s * ) in equation (5.20). (Online version in colour.) Definition 5.1. Using the exact circuit description, we compute a 2 × 2 operator H q defined as follows: H q acts on a two-dimensional Hilbert space corresponding to the low-energy subspace of the circuit Hamiltonian. The basis for H q in that subspace is chosen to diagonalize the low-energy projection ofφ. The energy levels of H q are chosen to exactly match the two levels of the circuit Hamiltonians, up to a constant shift. Once we obtain the relationship between b(s), f (s) and H q , we find b(s) and f (s) by requiring that where δ > 0 is a certain precision parameter we discuss below (ideally δ = 0).
Note that the true effective Hamiltonian H eff is isospectral to H q and is a rotation of H q to the basis determined by U eff , as will be prescribed in §6. In this section we only obtain explicit values of θ (the timescale in the error bounds (2.32)) for an evolution up to s = s * , and we demonstrate an improvement (small for CSFQ qubits and diverging as Θ(Λ) for CJJ qubits) over the JRS version, which yields

(i) Compound Josephson junction rf SQUID
Consider a D-wave (CJJ rf SQUID) qubit [31]. It consists of a large (main) loop and a small (CJJ) loop subjected to external flux biases Φ q x and Φ CJJ x , respectively. The CJJ loop is interrupted by two identical Josephson junctions connected in parallel with total capacitance C. For illustration purposes, we represent this loop as a single junction with some external phase control in a circuit diagram in figures 2 and 5. The two counter-circulating persistent current states along the main loop comprise the qubit |0 and |1 states, and can be understood as the states localized in the two wells of a double-well potential, described below.
The circuit Hamiltonian of this qubit can be written as in equation ( While H CJJ describes the physical circuit, we wish to implement the low-energy Hamiltonian of a qubit with frequency ω q , as defined by equation (5.19), using the approximate method given in definition 5.1. We now discuss how to make this transition. Treating the term E J b cos φ + E L (φ − f ) 2 as a classical potential in the variable φ, it represents a cosine potential superimposed on a parabolic well. The two lowest states in this potential are the qubit states, separated by ω q . These two states need to be separated from non-qubit states, and the corresponding gap is given by half the plasma frequency ω pl .
For a transmon, where E L = 0, one has ω q = ω pl − E C [50], where the plasma frequency is given by In the presence of the parabolic well, there are additional levels in local minima of the raised cosine potential. For f = 0, the two degenerate global minima appear at φ = ±π and the lowest local minima at φ = ±3π . Thus, to ensure that the additional levels in the local minima are higher than the qubit frequency, we can set min ω pl (s) = ω pl (0) ≈ (±3π ) 2 E L − (±π ) 2 E L = 8E L π 2 . Next, using b(0) = 1, if E C E J (as it must, to ensure ω q ω pl ) then E L = O( E C E J ) E J , which we will assume: We now wish to choose the controls of H CJJ so that H q in definition 5.1 takes the form H q (s) = ξ (s)X + ζ (s)Z, (5.24) so that ζ (s) = ω q s (compare to equation (5.19)). Focusing just on the minima at φ = ±π but now allowing f > 0, we have ζ (s) = E L (−π − f (s)) 2 − E L (π − f (s)) 2 , so that, upon neglecting the f 2 term, (5.25) subject to f (1) < π, i.e. we have the additional constraint ω q < 4E L π 2 .
Following [50], we can identify the bandwidth (peak-to-peak value for the charge dispersion of the energy levels in the periodic potential) of the E L = 0 Hamiltonian with the coefficient ξ (s) in the effective qubit Hamiltonian. Under the assumed inequality (5.23), equation (2.5) of [50] with m = 0 yields This mathematical model in fact describes a family of qubits, different by ω q , ω pl (1) and δ. The family is spanned by varying the ratio E J /E C and B, in the region where both are 1 to ensure the applicability of equation (5.26) and the smallness of the precision parameter δ. Note that in the E J /E C 1 and B 1 regime the aforementioned conditions ω q < ω pl and ω q < 4E L π 2 are automatically satisfied. Among the qubits in the family, a smaller ω q /ω pl (1) will allow a (relatively) faster anneal while the qubit approximation is maintained, but exactly how E J /E C and B (or equivalently ω q /ω pl (1) and δ) enter needs to be investigated via the adiabatic theorem, which we will delay until we analyse a simpler CSFQ case below.
We have thus shown how to reduce the circuit Hamiltonian H CJJ to an effective qubit Hamiltonian H q , and how the circuit control functions b(s) and f (s) relate to the effective qubit annealing schedule functions ξ (s) and ζ (s).

(ii) Capacitively shunted flux qubit
We now repeat the analysis for a periodic φ, i.e. for H CSFQ (equation (5.17b)). In this case, the potential E J b cos φ − E α cos((φ − f )/2) exhibits only two wells. For simplicity of the analysis, we instead choose to work with the Hamiltonian H CSFQ, sin given in equation (5.18). Recall that this Hamiltonian omits one of the terms in the trigonometric decomposition of cos((φ − f )/2) and has the benefit that the wells are centred exactly at φ = ±π for all f . Thus, it ignores the diabatic effects from the wells shifting along the φ-axis in the complete CSFQ Hamiltonian (5.17b). That effect can be included in the calculation straightforwardly, but for our example we choose the simplest non-trivial case. Each well independently experiences narrowing as b grows, leading to diabatic transitions out of the well's ground state. The physical meaning of the adiabatic timescale is to characterize the dynamics associated with this deformation of the harmonic oscillator, but by using the general machinery of our and the JRS bounds, we can obtain the result via algebra alone, without having to rely on physical intuition.
To apply the different versions of the adiabatic theorem expressed in corollary 2.2, we will need bounds on the derivatives of the simplified CSFQ Hamiltonian (5.18) (we drop the subscript and hat symbols for simplicity), and In the JRS case, one directly bounds the operator norm: and In the case of our new version of the adiabatic theorem, we will need bounds on the projected quantities. In any case, it is clear that we need to find bounds on the derivatives of b and f , which we now proceed to derive. The effective Hamiltonian. Define the well basis as the low-energy basis diagonalizing φ projected into the low-energy subspace. The qubit Hamiltonian in the well basis (see definition In the limit E α E J , we can approximate the width of the wells as equal, which leads to (in this case the same result is obtained with the complete potential E α cos((φ − f )/2)). We can also neglect the adjustment to the tunnelling amplitude through the barrier of height bE J coming from the bias ζ (s) ≤ 2E α between wells. This again uses E α E J . Repeating the argument leading to equation (5.26), the zero-bias expression (eqn (2.5) of Koch et al. [50] with m = 0) holds for the tunnelling amplitude, so we can reuse equation (5.26). This expression also uses E C E J . The more rigorous statement of the approximate equality in equation (5.26) is postulated in the conjecture below. In figure 6, we contrast the special regime of these approximations, which we call the well approximation, with the traditional schedule for quantum annealing.
Reducing the number of parameters. We choose the following notation for the ranges of b and f : In total, our CSFQ Hamiltonian has five parameters, E C , E J , E α , B and F , i.e. four dimensionless parameters since B and F are already dimensionless. We take E C to represent an overall energy scale and define the dimensionless parameter A as the ratio appearing in ξ (s), The parameter space can be reduced by setting F = π/3. Note that the maximum allowed F is π , at which f | s=1 required to fit the schedules will diverge. Making F really small just makes  We now make use of ω q = ξ (0) = ζ (1). This means that the annealing schedule is such that the start and end energy approximately coincide, as is traditional for the idealized qubit model of annealing (1 − s)X + sZ. This allows us to write i.e. the ratio E α /E C is also determined by A. Having fixed the dimensionless parameters E J /E C and E α /E C in terms of the single parameter A, and having fixed F at a numerical value, we are left only with A and B, i.e. we have reduced the original four dimensionless parameters to two. Let us now state the conjecture that replaces equation (5.26) by a rigorous statement.

Conjecture 5.2.
For a desired multiplicative precision , there exists a minimum A 0 ( ) such that for all A ≥ A 0 , The two derivatives ξ and ξ are also given by the derivatives of equation (5.33) to the same multiplicative precision .
The final transverse field needs to be negligible in quantum annealing. If our tolerance to a finite transverse field is δ, then let This implicitly defines B 0 (δ, A) > 1. So our two dimensionless parameters live in a range A ∈ [A 0 ( ), ∞] and B ∈ [B 0 (δ, A), ∞]. Their physical meaning is that A is the (root of the) area under the barrier in appropriate dimensionless units at the beginning of the anneal, and B is how much the barrier has been raised at the end relative to the beginning. We note that both B 0 and A 0 are rather large numbers for reasonable and δ, 11 so we intend to investigate the scaling of the adiabatic timescale θ in the limit A, B → ∞. The relationship between A and B as they approach that limit may be arbitrary; we do not make any additional assumptions about this. The gap to the non-qubit states is, to leading order, determined by the plasma frequency which is the same as equation (5.22) for the D-wave qubit. Even though ω pl (b) attains its minimum value at b(1) = 1, we will find that the terms in the numerator of the adiabatic theorem overwhelm it in such a way that only ω pl (B) at the end of the anneal matters.
Repeating the reasoning of the CJJ qubit case above, ξ (0) = ω q serves as the definition of ω q , and the time-dependent controls f (s) and b(s) should be (approximately, using equation (5.33)) chosen as ζ (s) Here δ B ≤ δ is the precision 12 we get for this choice of B. The quantity δ B and the ratio of the qubit frequency ω q = ξ (0) = E α (equation (5.34)) to the plasma frequency at the end of the anneal ω pl (B) = E C A √ B/8 are the two independent parameters we will use to present the final answer for θ new . The relationship of these two parameters with A and B is given by The derivatives b , b , f and f . First, from equation (5.38a) we have Since A 1 and b ≥ 1, we can neglect the subleading term 3/4 We do the same in the calculation of the second derivative: We will use a change of integration variable We also note that b and b are exponentially large in A( b(s) − 1), so they have the potential of becoming the leading terms in our estimate for the adiabatic timescale.
Completing the proof of the result claimed in equation (5.20). We show below that H does not grow with the cutoff Λ, so we apply corollary 2. ≈ ω pl /2, we have ⎞ ⎠ ds. (5.45) Returning to equation (5.28a), we now substitute the derivatives of b and f we found in terms of A and b, using equations (5.32), (5.34), (5.40a) and (5.42): and Here o(1) means going to zero in the limit A → ∞, or b → ∞. We will omit the (1 + o (1)) clause below when working with leading-order expressions. Let us substitute the expressions obtained so far into the integral I (equation (5.45)) and change variables to db using equation (5.44): (5.48) where we also used equation (5.37). The two terms depend on A and b in exactly the same way: The integral can be computed analytically in terms of the exponential integral function, but it is more insightful to observe that it is dominated by the upper integration limit, under the assumption that b(s * ) 1. Indeed, since B 1, there is a range of s * close to 1 for which equation (5.38b) gives b(s * ) 1. In that regime, The full bound for θ is therefore, using equations (5.37), (5.45), (5.47) and (5.51), . (5.54) The ratio of the qubit frequency to the gap is what one would intuitively expect from the adiabatic theorem, but the other factors can only be obtained after a detailed calculation such as the one performed here.
Completing the proof of the result claimed in equation (5.21). Since we have already shown that H does not grow with the cutoff Λ (equation (5.47a)), we now use equation (2.32b) (corollary 2.2) for the CSFQ Hamiltonian.
It turns out that there is no benefit from the projection in PH P , so we just use PH P ≤ H and focus on the off-diagonal terms PH Q and PH Q to obtain an improvement over the JRS bound (5.54). Starting from equation (5.27), we have Thus we need to estimate the leading order of the bound on P cosφQ and P sinφQ/2 . For this estimate, we make use of the well approximation: the eigenstates are approximately the states of a harmonic oscillator centred at each well ( figure 6). Indeed, recall that H CSFQ, sin (equation (5.18)) is a Hamiltonian representing a double-well potential centred exactly at φ = ±π for all f . We thus approximate H CSFQ, sin as the sum of Now P projects onto the span of the ground states of these two Hamiltonians, while Q projects onto the span of the first and higher excited states. Write δφ L,R ≡φ ± π ; then the expression for the position operators δφ in terms of the corresponding harmonic oscillator creation and annihilation operators is 13 We can now estimate where |g L,R are the ground states in the corresponding wells and we neglected the matrix elements of cos φ that mix the wells. We proceed as follows: . (5.59c) 13 To see this, consider the standard one-dimensional quantum harmonic oscillator Hamiltonian H = αp 2 + βx 2 , where α = 1/(2m) and β = mω 2 /2, which after the introduction of the standard creation and annihilation operators givesx It follows from equations (5.61a) and (5.62) that we may neglect PH (s)Q relative to H (s) . We may thus proceed from equation (5.49) but multiply the right-hand side by O(E C /(E J b(s))) 1/2 = O(1)(1/(A b(s))): where in the last approximate equality we applied the same reasoning as in equation (5.50).

(iii) Comparison of the two bounds for the CSFQ
To compare the two bounds, it is useful to express everything in terms of two parameters at s * only: 1 − s * + δ B and ω q /ω pl (b(s * )). Note that combining equations (5.34), (5.37) and (5.38b) gives . (5.70) Thus, since equation (5.67b) shows that the new bound is related to the JRS bound by the factor 1/(A b(s * )), using the new bound leads to a logarithmic correction of the original adiabatic timescale: We conclude that there are two competing small numbers, 1 − s * + δ B and ω q /ω pl (b(s * )). The gap to the third state should be much larger than the qubit frequency, i.e. ω pl (b(s)) ω q for all s. The expression 1 − s * + δ B (recall its definition in equation (5.38b)) times ω q can be interpreted as a residual transverse field h x at s = s * . This residual transverse field should satisfy h x /ω q = 1 − s * + δ B 1 in the regime where the expression θ (s * ) for the adiabatic timescale over the interval [0, s * ] is valid. Using equations (5.54) and (5.71) we may rewrite the two bounds as Thus, if the geometric mean h x ω pl ω q , then the effective dynamics stays within the qubit approximation well. Our new bound adds a logarithmic correction to this estimate and is tighter than the JRS bound since ω pl (b(s * )) > h x . Finally, we note that a brute-force calculation we present in appendix A yields an equivalent bound. (5.80)

Effective Hamiltonian
In this section, we will show that the effective evolution in a d P -dimensional low-energy subspace that is an image of P(s) is best described by a d P × d P effective Hamiltonian: This is written in the full Hilbert space even though we know that for all s > 0, P(s)|φ(s) = |φ(s) as long as the same holds for the initial state |φ 0 . This suggests that we could write the evolution as generated by a d P × d P matrix in the lowenergy subspace-the effective Hamiltonian. Of course, one can trivially do this by first undoing the evolution generated by U ad , i.e. by first changing the basis in a time-dependent manner via Let the eigenvectors of H(0) in the low-energy subspace be {|λ i } d P i=1 , and let the basis vectors defining the new d P -dimensional Hilbert space we map into be {|e i } d P i=1 . Then the isometry V 0 corresponding to the projection P 0 ≡ P(0) can be chosen as which is t f -dependent. where G = G(s) is a gauge (geometric connection) term in the generator for the basis change, which we assume to be block-diagonal (G = PGP + QGQ). We prove in appendix B that any such U G eff will satisfy the intertwining property much like equation (2.11) for U ad : We then let U G eff be our time-dependent change-of-basis transformation: Now, (∂/∂s)U G † eff = U G † eff (G † + [P, P ]) so that, using equation (6.2), we have where and we defined the time-dependent isometry into the effective basis at any s. Note that, by combining our notation, we can write |ψ(s) = V G |φ(s) and |φ(s) = V G † |ψ(s) . In practice, H eff and O eff can be found by truncation of the total Hilbert space to some large cutoff and working with truncated finite-dimensional matrices O, H, U and V. The error introduced by the cutoff may be estimated by trying several cutoffs and extrapolating. We defer a more rigorous treatment of this error to future work.
Let us now discuss the gauge G. There are two natural reasons for choosing G = 0. The first is that if we wish to keep the basis change (and thus the operators O G eff = V G (s)OV G † (s)) t findependent, then G itself must be t f -independent. Thus, by equation (6.14), the only choice that leads to t f -independent H G eff (s) is G = 0. The second is that the choice G = 0 is the one that minimizes the norm of the derivative of any observable. This can be interpreted as the desirable consequence of not imparting any additional geometric phases that artificially speed up the evolution of observables in the given observation frame. To show this explicitly, note first that since we assumed that G is block-diagonal, we cannot choose the block-off-diagonal form G = −[P , P] to cancel the time-dependence of the operators. When an operator X is block-diagonal so that in particular PXP = 0, then also V G XV G † = 0 since V G just maps onto the space the projector selects. With this, it is clear that since P[P, P ]P = 0, we have ∂ ∂s with the norm vanishing in general only when G = 0.

Conclusion
Starting with Kato's work in the 1950s, work on the adiabatic theorem of quantum mechanics has resulted in rigorous bounds on the convergence between the actual evolution and the approximate, adiabatic evolution. These bounds were initially derived for Hamiltonians with bounded-norm derivatives and then conjectured without presenting the explicit form for the unbounded case, subject to assumptions restricting the class of Hamiltonians to being 'admissible', which essentially meant that norms of certain functions of H and its derivatives were not allowed to diverge. In this work, we have obtained new bounds which are presented in explicit form and can be applied after the introduction of an appropriate cutoff to Hamiltonians whose derivatives are unbounded. After the cutoff all the derivatives are bounded by a function of the cutoff scale, but our bounds capture the physically relevant cases where the adiabatic timescale is independent of the cutoff. To achieve this, we introduced a different assumption, relating H to a power of H via a simple-to-check positivity condition (equation (2.26)). With this assumption, we derived a new form of the adiabatic theorem. We expect that this adiabatic theorem will prove to be useful in a variety of situations, e.g. in the context of adiabatic quantum computing using superconducting qubits or trapped ions, where the physical degrees of freedom correspond to (perturbed) harmonic oscillators. To demonstrate and illustrate the latter, we performed a calculation of the adiabatic timescale characterizing the accuracy of the qubit approximation of the circuit Hamiltonian of a capacitively shunted flux qubit. Specifically, we considered a time evolution fashioned after quantum annealing that attempts to reduce the qubit transverse field X linearly as (1 − s)X. The result shows that after some s * close to 1 the state generally escapes from the qubit approximation. Specifically, higher oscillator states become populated in each well. We do not expect this leakage effect to introduce a significant change in the outcome of a single-qubit quantum anneal, since the end-measurement is just a binary measurement of which well the flux is in, not the projection onto the lowest eigenstates. Thus, the non-qubit eigenstates become categorized as 0 or 1 depending on the sign of the flux. It remains an open question what the effect of this type of leakage is in the case of multi-qubit quantum dynamics, and whether it impacts the prospects of a quantum speed-up.
Data accessibility. This article has no additional data. The second term is subleading, so where in the second equality we used [P, G] = 0, which follows from G being block-diagonal (G = PGP + QGQ). Using the fact that P is block-off-diagonal (equations (2.13) and (2.14)), we simplify the last two terms as